I have an equation where there is a single summation on both sides of the equation.
In the image provided taken from the article [A], they did some simplification which I am unable to understand.
They used the result of 19 and substituted in the 16 to get equation 20.
My question is what happened to the summation on both sides?
[A] L.A. Bergman, J.K. Hall, G.G.G. Lueschen, D.M. McFarland: "Dynamic Green's functions for Levy plates", J. Sound Vib. 162-2 (1993), 281-310. doi:10.1006/jsvi.1993.1119
For simplicity, let $(16)$ be of the form of
$$\sum_{n=1}^\infty a_n \sin(n\pi x)=\delta(x-\xi)\delta(y-\eta)$$
and $(19)$ is of the form of
$$\delta(x-\xi) = \sum_{n=1}^\infty C_n \sin (n\pi x)$$
After they subsitute $(19)$ into $(16)$, since
$$\sum_{n=1}^\infty a_n \sin(n\pi x)=\sum_{n=1}^\infty C_n\delta(y-\eta)\sin(n\pi x)$$
Hence
$$\sum_{n=1}^\infty( a_n - C_n \delta(y-\eta)) \sin(n\pi x)=0$$
Hence the LHS is the fourier series of the zero function, hence
$$a_n = C_n \delta(y-\eta)$$